Define half life period and decay constant for a radioactive substance. Deduce the relation between them.

Explanation

Half life period (T) : Time taken for the number of radio active nuclei to disintegrate to half of its original number of nuclei is called Half life period.
Decay constant `(lambda)` : The ratio of the rate of radioactive decay to the number of nuclei present at that instant.
It is a proportional constant and is denoted by `'lambda'`.
`therefore = (-((dN)/(dt)))/(N)`
Relation between half the period and decay constant :
1. The radioactive law `N = N_(0)e^(-lambda t)` states that the number of radioactive nuclei in a radioactive sample decreases exponentially with time. Here `lambda` is called decay constant.
2. If `N_(0)` is the number of nuclei at t = 0 and N is the radioactive nuclei at any instant of time 't'.
3. Substituting `N=(N_(0))/(2)` at t = T in `N = N_(0) e^(-lambda t)`.
Where T is half life of the radioactive substance.
4. `(N_(0))/(2)=N_(0) e^(-lambda T)`
`e^(lambda T) = 2`
` lambda T = l n 2`
`T=(l n2)/(lambda)=(2.303log_(10)^(2))/(lambda)`
`therefore T = (0.693)/(lambda)`